Is there a materialist explanation of mathematics?

[tonyrey]

Probably. The material scent of three dog biscuits may be stronger than the scent of one.

In that case what the dog sees is not taken into account!

 

[tonyrey]

You don’t discover axioms. You invent a set of axioms then manipulate them to get theorems. A true theorem follows from the axioms, a false theorem doesn’t, but the axioms are always just inventions.

So the axioms don’t correspond to reality in any way?

 

[Tomdstone]

So the axioms don’t correspond to reality in any way?

Some axioms are self evident truths, but others may not be. Two things equal to the same thing are equal to one another is more or less self evident, but AB = BA is not. In any case, the axioms are a starting point for the mathematical system you are going to build.

 

[Charlemagne_III]

Some axioms are self evident truths, but others may not be. Two things equal to the same thing are equal to one another is more or less self evident, but AB = BA is not. In any case, the axioms are a starting point for the mathematical system you are going to build.

Can you give a mathematical equivalent of it being not an axiom that AB=BA?

 

[Tomdstone]

Can you give a mathematical equivalent of it being not an axiom that AB=BA?

In some mathematical systems the commutative law for multiplication will hold true, whereas in others not so.

 

[Charlemagne_III]

In some mathematical systems the commutative law for multiplication will hold true, whereas in others not so.

So in which mathematical system in particular would the commutative law for multiplication not be axiomatic?

 

[Tomdstone]

So in which mathematical system in particular would the commutative law for multiplication not be axiomatic?

Take the simple case of multiplication of 3x3 matrices.

 

[Charlemagne_III]

Take the simple case of multiplication of 3x3 matrices.

O.K. Well, I’m not getting it, so I won’t ask again. Thanks.

 

[Charlemagne_III]

I don’t expect anyone to see every philosophical problem the same way I do. But people who take philosophy seriously should at least be able to recognize where there ARE philosophical problems.

For example, I take it we both believe in free will. But there are numerous philosophical problems with it.

The philosophical problems are outweighed by the numerous arguments in favor of it, including the common sense argument…

The arguments against it come mostly from the materialists/determinists, who have problems of their own proving materialism/determinism.

Yes, you can take philosophy seriously and recognize there are problems with almost every major philosophical proposition. There will probably never be universal consensus on the solution to any philosophical proposition, as there will probably never be universal consensus on the true religion. We don’t expect everybody to agree with us. We just have lots of fun trying to find out if they will.

 

[Tomdstone]

O.K. Well, I’m not getting it, so I won’t ask again. Thanks.

Another simple way to prove non-commutativity is to let matrix A be a (4x3) matix which means that it has 4 rows and 3 colums; let the B matrix be a (3x4) matrix which means that it has 3 rows and 4 columns

Let us now multiply A * B
what we get is a (4x3) * (3x4) = (4x4) matrix as a result

If we now multiply B * A,
it is a (3x4) * (4x3) matix , which is a (3x3) matrix.

So A * B does not equal B * A.

Look up Matrix multiplication or non-commutativity of matrix multiplication.

 

[tonyrey]

Some axioms are self evident truths, but others may not be. Two things equal to the same thing are equal to one another is more or less self evident, but AB = BA is not. In any case, the axioms are a starting point for the mathematical system you are going to build.

What do the self-evident truths refer to?

 

[inocente]

I quoted up to and including the one statement that I want to examine:
“A true theorem follows from the axioms, a false theorem doesn’t.”

That statement is not clear enough for me to work with, so with your permission I would like to consider the following:

“Every true statement in number theory is a logical consequence of the axioms, and no false statement in number theory is a logical consequence of the axioms.”

Is that something that you might have asserted, or do you disagree with that statement?

I’ll quibble that we’ve both used the word “true”, and “true” isn’t absolute but relative to the rules of the particular system. So by standard definition:

An mathematics, the term axiom is used in two related but distinguishable senses: “logical axioms” and “non-logical axioms”. Logical axioms are usually statements that are taken to be true within the system of logic they define (e.g., (A and B) implies A), while non-logical axioms (e.g., a + b = b + a) are actually defining properties for the domain of a specific mathematical theory (such as arithmetic). - en.wikipedia.org/wiki/Axiom

A theorem is true if it conforms to the axioms:

In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems—and generally accepted statements, such as axioms. The proof of a mathematical theorem is a logical argument for the theorem statement given in accord with the rules of a deductive system. - en.wikipedia.org/wiki/Theorem

*In the meantime …

First,

Do you acknowledge that a conjecture can be discovered? For example, somebody considers adding consecutive odd numbers: 1+3 = 4, 1+3+5 = 9, 1+3+5+7 = 16, and notices that the results are squares. Quite quickly, a conjecture can be formulated. What is it that you believe prevents a generalization that is discovered via examples from being given the label “axiom”? After all, if you think that all axioms must be “justified logically” by deducing them from other axioms, then you will simply find yourself trapped in circular reasoning.*

No, theorems must be able to be traced back to the axioms, but the axioms don’t have to be true. For example:

*Euclidean geometry is an axiomatic system, in which all theorems (“true statements”) are derived from a small number of axioms:

…] “To draw a straight line from any point to any point.”
“To produce [extend] a finite straight line continuously in a straight line.”
“To describe a circle with any centre and distance [radius].”
“That all right angles are equal to one another.”
The parallel postulate: “That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.” - en.wikipedia.org/wiki/Euclidean_geometry#Axioms*

In non-Euclidean geometries, those axioms (especially the fifth) are modified or replaced, and there are different theorems as a result.

*Second,

What distinction between discovery and invention do you have in mind? Insofar as intellectual property rights are concerned, I believe it is an established matter of law that an axiom cannot be patented.*

I think there was the idea in ancient Greece that axioms should be self-evident, in which case they could be said to be discovered, but really they are invented. I’d say theorems are discovered, since they have to be found within the system.

 

[inocente]

There is a materialist basis for mathematics, since mathematics corresponds to the rules of nature, as in E= mc2.

Are you claiming that no Catholic can acknowledge that the physical law is mathematical without being called a materialist? You might want to look up the meaning of materialism. Also the difference between applied and pure math.

 

[tonyrey]

I’ll quibble that we’ve both used the word “true”, and “true” isn’t absolute but relative to the rules of the particular system. So by standard definition:

An mathematics, the term axiom is used in two related but distinguishable senses: “logical axioms” and “non-logical axioms”. Logical axioms are usually statements that are taken to be true within the system of logic they define (e.g., (A and B) implies A), while non-logical axioms (e.g., a + b = b + a) are actually defining properties for the domain of a specific mathematical theory (such as arithmetic). - en.wikipedia.org/wiki/Axiom

A theorem is true if it conforms to the axioms:

In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems—and generally accepted statements, such as axioms. The proof of a mathematical theorem is a logical argument for the theorem statement given in accord with the rules of a deductive system. - en.wikipedia.org/wiki/Theorem

No, theorems must be able to be traced back to the axioms, but the axioms don’t have to be true. For example:

*Euclidean geometry is an axiomatic system, in which all theorems (“true statements”) are derived from a small number of axioms:

…] “To draw a straight line from any point to any point.”
“To produce [extend] a finite straight line continuously in a straight line.”
“To describe a circle with any centre and distance [radius].”
“That all right angles are equal to one another.”
The parallel postulate: “That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.” - en.wikipedia.org/wiki/Euclidean_geometry#Axioms*

In non-Euclidean geometries, those axioms (especially the fifth) are modified or replaced, and there are different theorems as a result.

I think there was the idea in ancient Greece that axioms should be self-evident, in which case they could be said to be discovered, but really they are invented. I’d say theorems are discovered, since they have to be found within the system.

Are **all **axioms and systems arbitrary?

 

[Tomdstone]

There is a materialist basis for mathematics, since mathematics corresponds to the rules of nature, as in E= mc2.

In the case of E=mc2, I think it is more a case of nature corresponding to a mathematical equation.

 

[inocente]

In that case what the dog sees is not taken into account!

Some animals can count, and some are good at it. For instance, here’s a comparison of the counting abilities of college students and rhesus monkeys:

journals.plos.org/plosbiology/article?id=10.1371/journal.pbio.0050328

So the axioms don’t correspond to reality in any way?

They don’t have to, no of course not. Just as philosophers allow themselves to ask “what if” questions, so do mathematicians. They can, and did, invent the axiom i[sup]2[/sup] = -1 to see where it took them, and one resulting theorem is the beautiful Euler’s Identity:

e[sup]i π[/sup] + 1 = 0

(where e is the base of natural logarithms).

Is the square root of -1 part of reality? Who can say, but it later turned out to be very useful in electrics for instance.

 

[inocente]

What do the self-evident truths refer to?

For instance, the first four axioms in Euclidean geometry (see middle of post #92 above) were taken to be self-evident, although the Greeks were less convinced about the fifth.

Are **all **axioms and systems arbitrary?

Some systems are produced to be useful in the real world, for instance standard arithmetic grew out of real world needs. But the branch of math called pure math is purely abstract conceptualizing, although sometimes the systems it produces turn out later to have real-world applications.

 

[tonyrey]

For instance, the first four axioms in Euclidean geometry (see middle of post #92 above) were taken to be self-evident, although the Greeks were less convinced about the fifth.

What is the reason that they are self-evident? Is it because they refer to facts about material reality - like numbers?

Some systems are produced to be useful in the real world, for instance standard arithmetic grew out of real world needs. But the branch of math called pure math is purely abstract conceptualizing, although sometimes the systems it produces turn out later to have real-world applications.

By chance?

 

[PseuTonym]

I’ll quibble that we’ve both used the word “true”, and “true” isn’t absolute but relative to the rules of the particular system.

It is not a quibble. It is one of the sources of our disagreement with each other.

A theorem is true if it conforms to the axioms:

I disagree. The concepts “true” and “provable” are different. If we are talking about statements in number theory, then the concept of “true” has been assigned meaning prior to the construction of any particular system of axioms. Whether or not a given statement is provable depends on the particular system. If the system of axioms is inconsistent, then all statements are provable, but it does not follow that all statements are in fact both true and false. If the system of axioms is incomplete, then at least one true statement is not provable from those axioms.

Thus, rewriting your claim, I would say:
A statement is a theorem if it is provable from the axioms. Truth in number theory is a topic that goes beyond any particular system of axioms.

 

[inocente]

What is the reason that they are self-evident? Is it because they refer to facts about material reality - like numbers?

Philosophers disagree about what might be self-evident. Is 2 + 2 = 11 self-evident? Perhaps you’d say yes after I told you I’m counting in base 3, or perhaps you wouldn’t.

By chance?

For instance, a great deal of math has been developed for string theory, but string theory may well fail when it is tested and so be completely wrong. None of that math will then have been found to be part of reality.